Fourier Series for Periods Other Than
Figure 2:
Left: Plot of , periodic of period ,Right: Plot of the Fourier series expansion for .
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Examples:
- Let be periodic between and . (See Figure 2left). Find the s and s of its Fourier expansion. For ;
For the other s;
For the other s;
We then have
Figure 2right shows how the series approximates to the function when only two, four, or eight terms are used.
- Find the Fourier coefficients for on to ;
Because the definite integrals are nonzero only for odd values of , it simplifies to change the index of the summation. The Fourier series is then
Figure 3 shows how the series approximates the function when two, four, or eight terms are used.
Figure 3:
Plot of Fourier series for for .
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- Find the Fourier coefficients for
over the interval [-2, 2] if it is periodic of period 4. Equations 7 and 8 apply.
Figure 4 shows how the series approximates to the function when 40 terms are used.
Figure 4:
Plot of Fourier series for for .
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With MATLAB,
Cem Ozdogan
2010-12-29